Compactifying Hypertoric Manifolds via Symplectic Cutting

Structure Of This Presentation

• Delzant Polytopes and Toric Symplectic Manifolds

• Their Hypertoric Analogues

• Compactification via Symplectic Cutting

• Outlook

Delzant Polytopes

A Delzant polytope $$\Delta \subseteq \mathbb{R}^{n}$$ is a convex polytope satisfying:

• (simple); $$n$$ edges meet at each vertex;

• (rational); each edge meeting a vertex $$p$$ is of the form $$p + tu_{i}$$, $$t \geq 0$$, $$u_{i} \in \mathbb{Z}^{n}$$;

• (smooth); for each vertex, respective edge vectors $$u_{1}, \ldots, u_{n}$$, can be chosen to form a $$\mathbb{Z}$$-basis for $$\mathbb{Z}^{n}$$.

Each $$\Delta \subseteq \mathbb{R}^{n}$$ can be written as

$\Delta = \bigcap\limits_{i} \{ x \in (\mathbb{R}^{n})^{\ast}\, : \, \langle x ,\, u_{i} \rangle + \lambda_{i} \geq 0 \},\quad \lambda_{i} \in \mathbb{R},$ where $$u_{i} \in \mathbb{Z}^{n}$$ are the inward-pointing normals to the facets of $$\Delta$$.

Symplectic Toric Manifolds

Definition: A $$2n$$-dimensional symplectic toric manifold is a compact connected symplectic manifold $$(M^{2n},\omega)$$ with an effective Hamiltonian action of an $$n$$-torus $$T^{n}$$, with corresponding moment map $$\mu : M \rightarrow\operatorname{Lie}(T^{n})^{\ast} \cong (\mathbb{R}^{n})^{\ast}$$.

This definition is easier to elaborate upon with an example.

Example

$$T^{n}$$ acts on $$\mathbb{C}^{n}$$ diagonally:

$(t_{1}, \ldots, t_{n}) \cdot (z_{1}, \ldots, z_{n}) = (t_{1}z_{1}, \ldots, t_{n}z_{n}),$

Moment map for the action is: $$\mu : \mathbb{C}^{n} \longrightarrow\mathbb{R}^{n}$$,

$\mu(z) = \frac{1}{2} \sum\limits_{k = 1}^{n} |z_{k}|^{2}e_{k} \in \mathbb{R}^{n}.$

Abuse of notation: Identify $$(\mathbb{R}^{n})^{\ast} = \mathbb{R}^{n}$$ and omit constants for moment maps.

Toric Varieties

$$T^{n}$$ acts diagonally on $$\mathbb{C}^{n}$$ preserving the Kähler structure.

Let $$\{ u_{1}, \ldots, u_{n} \}$$ be inner-normals to some Delzant $$\Delta$$. They generate a Lie algebra $$\mathfrak{n}$$ for some sub-torus $$N \subseteq T^{n}$$.

$$\implies$$ exact sequences, where $$\pi: e_{i} \mapsto u_{i}$$:

$0 \longrightarrow\mathfrak{n} \overset{\imath}{\longrightarrow} \mathbb{R}^{n} \overset{\pi}{\longrightarrow} \mathbb{R}^{d} \longrightarrow 0,$

dualising

$0 \longleftarrow\mathfrak{n}^{\ast} \overset{\imath^{\ast}}{\longleftarrow} \mathbb{R}^{n} \overset{\pi^{\ast}}{\longleftarrow} \mathbb{R}^{d} \longleftarrow 0,$

or exponentiating

$1 \longrightarrow N \overset{\imath}{\longrightarrow} T^{n} \overset{\pi}{\longrightarrow} T^{d} \longrightarrow 1.$

$$N$$ acts on $$\mathbb{C}^{n}$$ via $$\imath$$, with moment map $\bar{\mu}(z) = (\imath^{\ast} \circ \mu)(z) = \frac{1}{2}\sum\limits_{k = 1}^{n} |z_{k}|^{2} \alpha_{k} \in \mathfrak{n}^{\ast},$

with $$\alpha_{k} = \imath^{\ast}(e_{k})$$.

If $$0$$ is a regular value for $$\bar{\mu}$$, then

$X = \bar{\mu}^{-1}(0)/N = (\imath^{\ast} \circ \mu)^{-1}(0)/N$

is a smooth Kähler quotient (assuming $$\{u_{1},\ldots,u_{n}\}$$ come from Delzant $$\Delta$$).

Convexity

Residual $$T^{d} = T^{n}/N$$ action on $$X$$, moment map $$\phi : X \rightarrow(\mathbb{R}^{d})^{\ast}$$.

For $$X$$ compact, Atiyah-Guillemin-Sternberg theorem $$\implies \operatorname{Im}(\phi)$$ is a convex polytope $$\Delta$$, and fixed-points of $$T^{d}$$ are its vertices.

Example

$0 \longrightarrow\mathfrak{n} \overset{\imath}{\longrightarrow} \mathbb{R}^{3} \overset{\pi}{\longrightarrow} \mathbb{R}^{2} \longrightarrow 0$

$u_{1} = (1,0),\ u_{2} = (0,1),\ u_{3} = (-1,-1),$ $\ker(\pi) = \langle e_{1} + e_{2} + e_{3} \rangle \subset \mathbb{R}^{3}$ $\implies \imath(t) = (t,t,t) \implies \imath^{\ast}(x,y,z) = x + y + z.$

$$T^{3}$$ on $$\mathbb{C}^{3}$$ moment map: $$\mu(z) = \tfrac{1}{2}\sum_{k = 1}^{3} |z_{k}|^{2}e_{k}$$, so $$N$$ moment map is:

$\bar{\mu}(z) = (\imath^{\ast}\circ \mu)(z) = \tfrac{1}{2}\|z\|^{2}$

$X = \mu^{-1}(c)/N = \{ \|z\|^{2} = 2c \}/N \cong S^{5}/S^{1} \cong \mathbb{C}\mathbb{P}^{2}.$

$X = \mu^{-1}(c)/N \cong \mathbb{C}\mathbb{P}^{2}$

has residual $$T^{2} = T^{3}/N$$ action:

$(t_{1}, t_{2}) \cdot [z_{0}: z_{1}: z_{2}] = [z_{0}:t_{1}z_{1}:t_{2}z_{2}].$

Moment map

$\phi(z) = \frac{1}{2}\bigg( \frac{|z_{1}|^{2}}{\|z\|^{2}},\, \frac{|z_{2}|^{2}}{\|z\|^{2}} \bigg),\quad \text{with } \operatorname{Im}(\phi) = \Delta_{2}.$

Fixed-points of $$T^{2}$$:

$\begin{array}{ccc} [1:0:0] & \mapsto & (0,0) \\ [0:1:0] & \mapsto & (1/2,0) \\ [0:0:1] & \mapsto & (0,1/2) \end{array}$

Hyperkähler Moment Maps

Analogous though now with $$\mathbb{H}^{n}$$.

Flat hyperkähler with three complex structures $$J_{1}, J_{2}$$, and $$J_{3}$$.

Fix $$J_{1}$$ so $$\mathbb{H}^{n} \cong T^{\ast}\mathbb{C}^{n}$$.

$$T^{n}$$-action on $$\mathbb{C}^{n}$$ induces $$T^{n}$$-action on $$T^{\ast}\mathbb{C}^{n}$$.

Hyperkähler moment maps

$\mu_{\mathbb{R}}(z,w) =\frac{1}{2}\sum_{k = 1}^{n}( |z_{k}|^{2} - |w_{k}|^{2} )e_{k} \in \mathbb{R}^{n},$

$\mu_{\mathbb{C}}(z,w) = \sum\limits_{k = 1}^{n}(z_{k}w_{k})e_{k} \in \mathbb{C}^{n}.$

Hypertoric Analogues

Choose $$\{u_{1}, \ldots, u_{n}\}$$ to get $$N \overset{\imath}{\hookrightarrow} T^{n}$$.

Mutatis mutandi, same construction as before:

$\bar{\mu}_{\mathbb{R}}(z,w) := (\imath^{\ast} \circ \mu_{\mathbb{R}})(z,w) = \tfrac{1}{2} \imath^{\ast}\bigg(\sum\limits_{k = 1}^{n}\big( |z_{k}|^{2} - |w_{k}|^{2} \big) e_{k} \bigg),$

$\bar{\mu}_{\mathbb{C}}(z,w) := (\imath_{\mathbb{C}}^{\ast} \circ \mu_{\mathbb{C}})(z,w) = \imath^{\ast}_{\mathbb{C}}\bigg( \sum\limits_{k = 1}^{n}(z_{k}w_{k}) e_{k} \bigg).$

Hyperkähler analogue $$M$$ to the Kähler quotient $$X$$ is

$M := \big( \bar{\mu}_{\mathbb{R}}^{-1}(\lambda) \cap \bar{\mu}_{\mathbb{C}}^{-1}(0) \big) / N.$

Hyperplane Arrangements

Residual $$T^{d} = T^{n}/N$$-action on $$M$$; has hyperkähler moment maps

$\phi_{\mathbb{R}}[z,w] = \tfrac{1}{2} \sum\limits_{k = 1}^{n}( |z_{k}|^{2} - |w_{k}|^{2} - \lambda_{k} )\alpha_{k},$

$\phi_{\mathbb{C}}[z,w] = \sum\limits_{k = 1}^{n} (z_{k}w_{k}) \alpha_{k}.$

Image $$\operatorname{Im}(\phi_{\mathbb{R}}) \subseteq \mathbb{R}^{d}$$ decomposes into a hyperplane arrangement: for $$y \in \mathbb{R}^{d}$$,

$F_{i} = \{ y \cdot u_{i} + \lambda_{i} \geq 0 \},\quad G_{i} = \{ y \cdot u_{i} + \lambda_{i} \leq 0 \},$

$H_{i} = F_{i} \cap G_{i}.$

Example - Hypertoric Analogue for $$\mathbb{C}\mathbb{P}^{2}$$

Extend $$T^{3}$$ diagonal action on $$\mathbb{C}^{3}$$ to $$T^{\ast}\mathbb{C}^{3}$$; now $$N$$ acts as $$t \cdot (z,w) = (tz,t^{-1}w)$$.

Hyperkähler quotient

$M = \Big( \bar{\mu}_{\mathbb{R}}^{-1}(\lambda) \cap \bar{\mu}_{\mathbb{C}}^{-1}(0) \Big) / N \cong T^{\ast}\mathbb{C}\mathbb{P}^{2}$

has residual $$T^{2}$$-action.

$\phi_{\mathbb{R}}[z,w] = \tfrac{1}{2}\sum\limits_{k=1}^{3}(|z_{k}|^{2} - |w_{k}|^{2}) - \lambda_{3}.$

The $$\{ H_{i} \}$$ divide $$(\mathbb{R}^{d})^{\ast}$$ into a union of closed convex polyhedra

$\Delta_{A} = \bigcap\limits_{i \in A} F_{i} \cap \bigcap\limits_{i \not\in A} G_{i}.$

Set $$\mathcal{E} := \phi_{\mathbb{C}}^{-1}(0) = \{ [z,w] \in M : z_{i}w_{i} = 0,\, \text{for all } i \} \subseteq M$$, which further decomposes

$\mathcal{E}_{A} := \{ w_{i} = 0 \text{ for all } i \in A, \text{ and } z_{i} = 0 \text{ for all } i \not\in A \},$ for subsets $$A \subseteq \{1,\ldots,n\}$$.

Lemma: If $$w_{i} = 0$$ then $$\phi_{\mathbb{R}}[z,w] \in F_{i}$$, and if $$z_{i} = 0$$, then $$\phi_{\mathbb{R}}[z,w] \in G_{i}$$.

The Core and the Extended Core of $$M$$

Call $$\mathcal{E}_{A} = \phi_{\mathbb{C}}^{-1}(0)$$ the extended core of $$M$$:

Each $$\mathcal{E}_{A} \subseteq M$$ is a $$d$$-dimensional Kähler subvariety with effective Hamiltonian $$T^{d}$$-action.

Lemma: $$\phi_{\mathbb{R}}(\mathcal{E}_{A}) \cong \bigcap\limits_{i \in A} F_{i} \cap \bigcap\limits_{i \not\in A} G_{i} =: \Delta_{A}$$, and $$\Delta_{A}$$ is corresponding Delzant polytope to $$\mathcal{E}_{A}$$.

We call $$\mathcal{L} := \bigcup\limits_{\Delta_{A} \text{ bounded}} \mathcal{E}_{A}$$, the core of $$M$$.

Residual $$S^{1}$$-Action

Additional $$S^{1}$$-action on $$T^{\ast}\mathbb{C}^{n}$$:

$\tau \cdot (z,w) = (z,\tau w).$

Descends to $$M$$, but only preserves $$J_{1}$$ structure, not $$J_{2}$$ nor $$J_{3}$$.

Does not act on $$M$$ as a sub-torus of $$T^{d}$$, but does when restricted to each $$\mathcal{E}_{A}$$.

For $$[z,w] \in \mathcal{E}_{A}$$,

$[z;\tau_{1} w_{1}, \ldots, \tau_{n} w_{n}] = [\tau_{1}^{-1}z_{1} , \ldots, \tau_{n}^{-1}z_{n}; \tau_{1} w_{1}, \ldots, \tau_{n}w_{n}],$

$\text{where } \tau_{i} = \begin{cases} \tau, \quad & \text{if } i \in A, \\ 1, \quad & \text{if } i \not\in A. \end{cases}$

Symplectic Cut

$$S^{1}$$-action on $$M$$ has proper moment map $$\Phi[z,w] = \tfrac{1}{2}\|w\|^{2}$$.

Extend it to $$M \times \mathbb{C}$$ via

$e^{i\theta} \cdot (m,\xi) = ( e^{i\theta} \cdot m, e^{i\theta} \xi),$ with moment map

$\rho_{\text{cut}} : M \times \mathbb{C}\rightarrow\mathbb{R}; \quad (m,\xi) \mapsto \Phi(m) + \tfrac{1}{2}|\xi|^{2}.$

The symplectic cut is the quotient

$M_{\epsilon-\text{cut}} := \rho_{\text{cut}}^{-1}(\epsilon)/S^{1} \cong \{m \in M : \Phi(m) < \epsilon\} \sqcup (\Phi^{-1}(\epsilon)/S^{1}).$

Compactification of $$M$$

$$S^{1}$$ acts on $$M$$, depending combinatorially on $$A \subseteq \{1,\ldots, n\}$$. Recall: $S^{1}_{A} = (\tau_{1}, \ldots, \tau_{n}), \text{ with } \tau_{i} = \tau\text{ if } i \in A; \tau_{i} = 1 \text{ otherwise}.$

Action comes from inclusion $$S^{1}_{A} \hookrightarrow T^{n} \rightarrow T^{d}$$, with moment map

$[z,w] \mapsto \Big\langle \phi_{\mathbb{R}}[z,w], \sum_{i \in A} u_{i} \Big\rangle.$

Cutting introduces new half-spaces

$\Delta_{A}^{(\epsilon)} := \Delta_{A} \cap \bigg\{ y \in \Delta_{A} : \Big\langle y,\, \sum_{i \in A} u_{i} \Big\rangle + \epsilon\geq 0 \bigg\}.$

Outlook From Here

• Hypertoric manifolds with non-compact cores;
• Applying localisation formulae;
• Geometric quantisation and lattice point enumeration?

References

• Delzant, T. – Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France, 116, 1988.

• Bielawski, R. and Dancer, A. – The geometry and topology of toric hyperkähler manifolds, Comm. Anal. Geom., 8, 2000.

• Proudfoot, N. – Hyperkahler analogues of Kahler quotients, Ph.D. Thesis, Harvard University, 2004.

• Lerman, E. – Symplectic Cuts, Math. Res. Lett., 2, 1995.